Section 6.3 Apparent Forces in Circular Motion © 2015 Pearson Education, Inc.

Section 6.3 Apparent Forces in Circular Motion

© 2015 Pearson Education, Inc.

2

Centrifugal Force?

• If you are a passenger in a car that turns a corner quickly, it is the force of the car door, pushing inward toward the center of the curve, that causes you to turn the corner.

• What you feel is your body trying to move ahead in a straight line as outside forces (the door) act to turn you in a circle.

A centrifugal force will never appear on a free-body diagram

and never be included in Newton’s laws.


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Apparent Weight in Circular Motion


4


• The force you feel, your apparent weight, is the magnitude of the contact force that supports you.

• When the passenger on the roller coaster is at the bottom of the loop:

• The net force points upward, so n > w.

• Her apparent weight is wapp= n, so her apparent weight is greater than her true weight.


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• Newton’s second law for the passenger at the bottom of the circle is

• From this equation, the passenger’s apparent weight is

• Her apparent weight at the bottom is greater than her true weight, w.


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• Newton’s second law for the passenger at the top of the circle is

• Note that wx is now positive because the x-axis is directed downward. We can solve for the passenger’s apparent weight:

• If v is sufficiently large, her apparent weight can exceed the true weight.


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• As the car goes slower there comes a point where n becomes zero:

• The speed for which n = 0 is called the critical speed vc. Because for n to be zero we must have , the critical speed is

• The critical speed is the slowest speed at which the car can complete the circle.


8

Example Problem

A handful of professional skaters have taken a skateboard through an inverted loop in a full pipe. For a typical pipe with a diameter 14 feet, what is the minimum speed the skater must have at the very top of the loop?


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A physics textbook swings back and forth as a pendulum. Which is the correct free-body diagram when the book is at the bottom and moving to the right?

QuickCheck 6.10

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Centripetal acceleration requires an upward force.

C.


A physics textbook swings back and forth as a pendulum. Which is the correct free-body diagram when the book is at the bottom and moving to the right?

QuickCheck 6.10

11

A car that’s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant?

QuickCheck 6.11


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QuickCheck 6.11


A.

A car that’s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant?

Now the centripetal acceleration points down.

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QuickCheck 6.12

A roller coaster car does a loop-the-loop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected.


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QuickCheck 6.12

A roller coaster car does a loop-the-loop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected.


The track is above the car, so the normal force of the track pushes down.E.

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Centrifuges


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Try It Yourself: Human Centrifuge

If you spin your arm rapidly in a vertical circle, the motion will produce an effect like that in a centrifuge. The motion will assist outbound blood flow in your arteries and retard inbound blood flow in your veins. There will be a buildup of fluid in your hand that you will be able to see (and feel!) quite easily.


17

An 18-cm-diameter ultracentrifuge produces an extraordinarily large centripetal acceleration of 250,000g, where g is the free-fall acceleration due to gravity. What is its frequency in rpm? What is the apparent weight of a sample with a mass of 0.0030 kg?

PREPARE The acceleration in SI units is

The radius is half the diameter, or r = 9.0 cm = 0.090 m.

SOLVE We can rearrange Equation 6.5 to find the frequency given the centripetal acceleration:

Example 6.10 Analyzing the ultracentrifuge


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Example 6.10 Analyzing the ultracentrifuge (cont.)

Converting to rpm, we find

The acceleration is so high that every force is negligible except for the force that provides the centripetal acceleration. The net force is simply equal to the inward force, which is also the sample’s apparent weight:

The 3 gram sample has an effective weight of about 1700 pounds!


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Example 6.10 Analyzing the ultracentrifuge (cont.)

ASSESS Because the acceleration is 250,000g, the apparent weight is 250,000 times the actual weight. This makes sense, as does the fact that we calculated a very high frequency, which is necessary to give the large acceleration.


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A coin sits on a turntable as the table steadily rotates counterclockwise. What force or forces act in the plane of the turntable?

QuickCheck 6.13


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A coin sits on a turntable as the table steadily rotates counterclockwise. What force or forces act in the plane of the turntable?

QuickCheck 6.13


A.

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A coin sits on a turntable as the table steadily rotates counterclockwise. The free-body diagrams below show the coin from behind, moving away from you. Which is the correct diagram?

QuickCheck 6.14


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C.

A coin sits on a turntable as the table steadily rotates counterclockwise. The free-body diagrams below show the coin from behind, moving away from you. Which is the correct diagram?

QuickCheck 6.14


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A car turns a corner on a banked road. Which of the diagrams could be the car’s free-body diagram?

QuickCheck 6.15


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QuickCheck 6.15

A car turns a corner on a banked road. Which of the diagrams could be the car’s free-body diagram?


E.

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Example Problem

A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure. When the car is at the highest point of the hill,

A. What is the force of gravity on the car?

B. What is the normal force of the road on the car at this point?


Section 6.4 Circular Orbits and Weightlessness


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Orbital Motion

The force of gravity on a projectile is directed toward the center of the earth.


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Orbital Motion

• If the launch speed of a projectile is sufficiently large, there comes a point at which the curve of the trajectory and the curve of the earth are parallel.

• Such a closed trajectory is called an orbit.

• An orbiting projectile is in free fall.


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Orbital Motion

• The force of gravity is the force that causes the centripetal acceleration of an orbiting object:

• An object moving in a circle of radius r at speed vorbit will have this centripetal acceleration if

• That is, if an object moves parallel to the surface with the speed


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Orbital Motion

• The orbital speed of a projectile just skimming the surface of a smooth, airless earth is

• We can use vorbit to calculate the period of the satellite’s orbit:


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Weightlessness in Orbit

• Astronauts and their spacecraft are in free fall.


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QuickCheck 6.19

Astronauts on the International Space Station are weightless because

A. There’s no gravity in outer space.

B. The net force on them is zero.

C. The centrifugal force balances the gravitational force.

D. g is very small, although not zero.

E. They are in free fall.


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QuickCheck 6.19

Astronauts on the International Space Station are weightless because

A. There’s no gravity in outer space.

B. The net force on them is zero.

C. The centrifugal force balances the gravitational force.

D. g is very small, although not zero.

E. They are in free fall.


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The Orbit of the Moon

• The moon, like all satellites, is simply “falling” around the earth.

• If we use the distance to the moon, r = 3.84 × 108 m, in:

we get a period of approximately 11 hours instead of one month.

• This is because the magnitude of the force of gravity, and thus the size of g, decreases with increasing distance from the earth.


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Example Problem

Phobos is one of two small moons that orbit Mars. Phobos is a very small moon, and has correspondingly small gravity—it varies, but a typical value is about 6 mm/s2. Phobos isn’t quite round, but it has an average radius of about 11 km. What would be the orbital speed around Phobos, assuming it was round with gravity and radius as noted?


Section 6.5 Newton’s Law of Gravity


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Gravity Obeys an Inverse-Square Law

• Gravity is a universal force that affects all objects in the universe.

• Newton proposed that the force of gravity has the following properties:

1. The force is inversely proportional to the square of the distance between the objects.

2. The force is directly proportional to the product of the masses of the two objects.


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Gravity Obeys an Inverse-Square Law

• Newton’s law of gravity is an inverse-square law.

• Doubling the distance between two masses causes the force between them to decrease by a factor of 4.


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Conceptual Example 6.11 Varying gravitational force

The gravitational force between two giant lead spheres is 0.010 N when the centers of the spheres are 20 m apart. What is the distance between their centers when the gravitational force between them is 0.160 N?

REASON We can solve this problem without knowing the masses of the two spheres. The key is to consider the ratios of forces and distances. Gravity is an inverse-square relationship; the force is related to the inverse square of the distance. The force increases by a factor of (0.160 N)/(0.010 N) = 16, so the distance must decrease by a factor of = 4. The distance is thus (20 m)/4 = 5.0 m.

ASSESS This type of ratio reasoning is a very good way to get a quick handle on the solution to a problem.


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Example 6.12 Gravitational force between two people

You are seated in your physics class next to another student 0.60 m away. Estimate the magnitude of the gravitational force between you. Assume that you each have a mass of 65 kg.

PREPARE We will model each of you as a sphere. This is not a particularly good model, but it will do for making an estimate. We will take the 0.60 m as the distance between your centers.


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Example 6.12 Gravitational force between two people (cont.)

SOLVE The gravitational force is given by Equation 6.15:

ASSESS The force is quite small, roughly the weight of one hair on your head. This seems reasonable; you don’t normally sense this attractive force!


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The force of Planet Y on Planet X is ___ the magnitude of .

A. One quarter

B. One half

C. The same as

D. Twice

E. Four times

QuickCheck 6.16


2M M

Planet X Planet Y

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The force of Planet Y on Planet X is ___ the magnitude of .

A. One quarter

B. One half

C. The same as

D. Twice

E. Four times

QuickCheck 6.16


2M M

Planet X Planet YNewton’s third law

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QuickCheck 6.17

The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled?

A. 250,000 N

B. 500,000 N

C. 1,000,000 N

D. 2,000,000 N

E. 4,000,000 N


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QuickCheck 6.17

The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled?

A. 250,000 N

B. 500,000 N

C. 1,000,000 N

D. 2,000,000 N

E. 4,000,000 N


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• If you traveled to another planet, your mass would be the same but your weight would vary. The weight of a mass m on the moon is given by

• Using Newton’s law of gravity (Eq. (6.15)) the weight is given by

• Since these are two expressions for the same force, they are equal and

Gravity on Other Worlds


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Gravity on Other Worlds

• If we use values for the mass and the radius of the moon, we compute gmoon = 1.62 m/s2.

• A 70-kg astronaut wearing an 80-kg spacesuit would weigh more than 330 lb on the earth but only 54 lb on the moon.


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QuickCheck 6.18

Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y

A. g = 2 m/s2

B. g = 4 m/s2

C. g = 8 m/s2

D. g = 16 m/s2

E. g = 32 m/s2


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QuickCheck 6.18

Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y

A. g = 2 m/s2

B. g = 4 m/s2

C. g = 8 m/s2

D. g = 16 m/s2

E. g = 32 m/s2


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QuickCheck 6.22

A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest?


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QuickCheck 6.22

A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest?


A

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Example 6.14 Finding the speed to orbit Deimos

Mars has two moons, each much smaller than the earth’s moon. The smaller of these two bodies, Deimos, isn’t quite spherical, but we can model it as a sphere of radius 6.3 km. Its mass is 1.8 × 1015 kg. At what speed would a projectile move in a very low orbit around Deimos?


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Example 6.14 Finding the speed to orbit Deimos (cont.)

SOLVE The free-fall acceleration at the surface of Deimos is small:


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Example 6.14 Finding the speed to orbit Deimos (cont.)

Given this, we can use Equation 6.13 to calculate the orbital speed:

ASSESS This is quite slow. With a good jump, you could easily launch yourself into an orbit around Deimos!


56

Example Problem

A typical bowling ball is spherical, weighs 16 pounds, and has a diameter of 8.5 in. Suppose two bowling balls are right next to each other in the rack. What is the gravitational force between the two—magnitude and direction?


Section 6.6 Gravity and Orbits


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Gravity and Orbits

• Newton’s second law tells us that FM on m = ma, where FM on m is the gravitational force of the large body on the satellite and a is the satellite’s acceleration.

• Because it’s moving in a circular orbit, Newton’s second law gives


59

Gravity and Orbits

• A satellite must have this specific speed in order to maintain a circular orbit of radius r about the larger mass M.


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Gravity and Orbits

• For a planet orbiting the sun, the period T is the time to complete one full orbit. The relationship among speed, radius, and period is the same as for any circular motion:

v = 2πr/T

• Combining this with the value of v for a circular orbit from Equation 6.21 gives

• If we square both sides and rearrange, we find the period of a satellite:


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QuickCheck 6.20

Two satellites have circular orbits with the same radius. Which has a higher speed?

A. The one with more mass.

B. The one with less mass.

C. They have the same speed.


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QuickCheck 6.20

Two satellites have circular orbits with the same radius. Which has a higher speed?

A. The one with more mass.

B. The one with less mass.



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QuickCheck 6.21

Two identical satellites have different circular orbits. Which has a higher speed?

A. The one in the larger orbit

B. The one in the smaller orbit



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QuickCheck 6.21

Two identical satellites have different circular orbits. Which has a higher speed?

A. The one in the larger orbit

B. The one in the smaller orbit



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QuickCheck 6.23

A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases?

A. Speed

B. Angular speed

C. Period

D. Centripetal acceleration

E. Gravitational force of the earth


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QuickCheck 6.23

A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases?

A. Speed

B. Angular speed

C. Period

D. Centripetal acceleration

E. Gravitational force of the earth


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Example 6.15 Locating a geostationary satellite

Communication satellites appear to “hover” over one point on the earth’s equator. A satellite that appears to remain stationary as the earth rotates is said to be in a geostationary orbit. What is the radius of the orbit of such a satellite?

PREPARE For the satellite to remain stationary with respect to the earth, the satellite’s orbital period must be 24 hours; in seconds this is T = 8.64 × 104 s.


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Example 6.15 Locating a geostationary satellite (cont.)

SOLVE We solve for the radius of the orbit by rearranging Equation 6.22. The mass at the center of the orbit is the earth:


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Example 6.15 Locating a geostationary satellite (cont.)

ASSESS This is a high orbit, and the radius is about 7 times the radius of the earth. Recall that the radius of the International Space Station’s orbit is only about 5% larger than that of the earth.


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Gravity on a Grand Scale

• No matter how far apart two objects may be, there is a gravitational attraction between them.

• Galaxies are held together by gravity.

• All of the stars in a galaxy are different distances from the galaxy’s center, and so orbit with different periods.


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Example Problem

Phobos is the closer of Mars’s two small moons, orbiting at 9400 km from the center of Mars, a planet of mass 6.4 × 1023 kg. What is Phobos’s orbital period? How does this compare to the length of the Martian day, which is just shy of 25 hours?


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Summary: General Principles


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Summary: General Principles


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Summary: Important Concepts


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Summary: Important Concepts


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Summary: Applications


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Summary: Applications


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Summary


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Summary


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Summary


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Section 6.3 Apparent Forces in Circular Motion © 2015 Pearson Education, Inc.

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Transcript of Section 6.3 Apparent Forces in Circular Motion © 2015 Pearson Education, Inc.